Discrete self-similar and ergodic Markov chains

TL;DR

This paper introduces a new class of discrete self-similar Markov chains on non-negative integers, explores their connection to self-similar processes, and analyzes their ergodic properties and spectral characteristics.

Contribution

It defines discrete self-similar Markov chains, establishes their relation to spectrally negative self-similar processes, and studies their ergodic behavior and spectral properties.

Findings

Chains are upward skip-free due to invariance property.

Scaled chains converge to self-similar processes.

Ergodic chains exhibit specific spectral and hypercontractivity properties.

Abstract

The first aim of this paper is to introduce a class of Markov chains on $\mathbb{Z}_+$ which are discrete self-similar in the sense that their semigroups satisfy an invariance property expressed in terms of a discrete random dilation operator. After showing that this latter property requires the chains to be upward skip-free, we first establish a gateway relation, a concept introduced in [26], between the semigroup of such chains and the one of spectrally negative self-similar Markov processes on $\mathbb{R}_+$. As a by-product, we prove that each of these Markov chains, after an appropriate scaling, converge in the Skorohod metric, to the associated self-similar Markov process. By a linear perturbation of the generator of these Markov chains, we obtain a class of ergodic Markov chains, which are non-reversible. By means of intertwining and interweaving relations, where the latter was recently introduced in [27], we derive several deep analytical properties of such ergodic chains including the description of the spectrum, the spectral expansion of their semigroups, the study of their convergence to equilibrium in the $\Phi$-entropy sense as well as their hypercontractivity property.